Optimal. Leaf size=58 \[ a x-\frac {b c n x^{n+1} \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );c^2 x^{2 n}\right )}{n+1}+b x \tanh ^{-1}\left (c x^n\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6091, 364} \[ a x-\frac {b c n x^{n+1} \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );c^2 x^{2 n}\right )}{n+1}+b x \tanh ^{-1}\left (c x^n\right ) \]
Antiderivative was successfully verified.
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Rule 364
Rule 6091
Rubi steps
\begin {align*} \int \left (a+b \tanh ^{-1}\left (c x^n\right )\right ) \, dx &=a x+b \int \tanh ^{-1}\left (c x^n\right ) \, dx\\ &=a x+b x \tanh ^{-1}\left (c x^n\right )-(b c n) \int \frac {x^n}{1-c^2 x^{2 n}} \, dx\\ &=a x+b x \tanh ^{-1}\left (c x^n\right )-\frac {b c n x^{1+n} \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );c^2 x^{2 n}\right )}{1+n}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 58, normalized size = 1.00 \[ a x-\frac {b c n x^{n+1} \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );c^2 x^{2 n}\right )}{n+1}+b x \tanh ^{-1}\left (c x^n\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b \operatorname {artanh}\left (c x^{n}\right ) + a, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int b \operatorname {artanh}\left (c x^{n}\right ) + a\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int a +b \arctanh \left (c \,x^{n}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (n \int \frac {1}{c x^{n} + 1}\,{d x} + n \int \frac {1}{c x^{n} - 1}\,{d x} + x \log \left (c x^{n} + 1\right ) - x \log \left (-c x^{n} + 1\right )\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int a+b\,\mathrm {atanh}\left (c\,x^n\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {atanh}{\left (c x^{n} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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